Unconventional reservoirs often have a low-permeability rock matrix that impedes fluid flow, making it difficult to extract hydrocarbons (or other fluids of interest) at commercially-feasible rates and volumes. Fortunately, various treatments can be used to increase the effective permeability of the formation. For example, when a proper acidic solution is injected into the formation, it dissolves regions of the matrix around the pores to create “wormholes” through which fluids can more easily flow.
To determine the desirability of an acidizing treatment, reservoir engineers may employ formation models with three spatial dimensions for simulating this unsteady process, numerically solving the fluid flow governing equations to determine the transient formation fluids' transient behavior before the treatment, as well as to determine the flow of injected treatment fluids for estimating changes to effective permeability in all directions, and to simulate the flow of formation fluids after the treatment while accounting for anisotropic permeability. See, e.g., Bagheri, M. and Settari, A., “Methods of Modelling full Tensor Permeability in Reservoir Simulators”, PETSOC-07-03-02, 2007; and Yetkin C., Ramiraz B., Al-Kobaisi, M., Ozkan, E., “A Simple Method to Account for Permeability Anisotropy in Reservoir Models and Multi-Well Pressure Interference Tests”, SPE 122972, 2009. Using such simulators, the treatment design engineers can then compare fluid flows “before” and “after” the treatment operation to evaluate and optimize the effectiveness of the treatment.
Unfortunately, such three spatial dimensional modeling often imposes computational resource requirements that are prohibitive and typically not justified in view of the limited amount of information regarding downhole reservoir conditions. Accordingly, various alternative approaches have been sought to significantly reduce the computational resource requirements while still providing sufficiently accurate results that account for, among other things, the effects of anisotropic permeability. See, e.g., Cline, S. B. and Tiab, D., “Studies in Vertical and Horizontal Well-Flow Behavior in cases of Permeability Anisotropy”, SPE 71085, 2001; and Azom, P. N., Srinivasan, S., “Modeling the Effect of Permeability Anisotropy on the Steam-Assisted Gravity Drainage (SAGD) Process”, SPE 149274, 2011. The accuracy of these alternative approaches remains insufficient for many applications.
It should be understood, however, that the specific embodiments given in the drawings and detailed description do not limit the disclosure. On the contrary, they provide the foundation for one of ordinary skill to discern the alternative forms, equivalents, and modifications that are encompassed together with one or more of the given embodiments in the scope of the appended claims.